Download Conditional Statements - Suffolk Public Schools Blog

LOGIC, CONDITIONAL
STATEMENTS AND
DEDUCTIVE
REASONING
Sol: G.1 b, c
Sec: 2.1-2.2
A conjecture is an educated guess based on known
information.
A statement is a sentence that is either true or false but
not both. Often represented by a letter such a p, q or r.
The Truth value is the truth or falsity of a statement.
NEGATIONS
The negation of a statement says it has the
opposite meaning.
So the negation would be
~p or ~q
read “not p or not q”
Example:
p: Suffolk is a city in Virginia.
The negation would be:
~p: Suffolk is not a city in Virginia.
COMPOUND STATEMENT
A compound statement is two or more statements
that are joined together.
Example:
p: Richmond is a city in Virginia.
q: Richmond is the capital of Virginia.
p and q:
Richmond is a city in Virginia and Richmond is the
capital of Virginia.
Two types – conjunction and disjunction
CONJUNCTION
A conjunction is a compound statement formed by
joining two or more statements with the word
“AND”.
Symbolic representation:  and
pq
“read” p and q
* A conjunction is true IFF both statements are
true.*
EXAMPLE:
USE THE FOLLOWING STATEMENTS TO WRITE A COMPOUND
STATEMENT FOR EACH CONJUNCTION THEN FIND IT’S TRUTH
VALUE.
p: One foot is 14 inches.
q: September has 30 days
r: A plane is defined by 3 non-collinear points.
a) p and q
One foot is 14 inches and September has 30 days.
b) r  p
A plane is defined by 3 non-collinear points and one foot is 14
inches.
c) ~ q  r
September does not have 30 days and a plane is defined by 3
non-collinear points.
d) ~ p  r
One foot does not have 14 inches and a plane is defined by 3
non-collinear points.
WRITE A COMPOUND STATEMENT FOR THE
CONJUNCTION AND FIND ITS TRUTH VALUE.
p: an elephant is a mammal
q: a square has four right angles
DISJUNCTION
A disjunction is a compound statement that joins
two or more statements with the word “or”.
Symbolic representation: p  q
“read” p or q
*A disjunction is true if at least one of the
statements are true.*
EXAMPLE:
USE THE FOLLOWING STATEMENTS TO WRITE A COMPOUND
STATEMENT FOR EACH DISJUNCTION THEN FIND IT’S TRUTH
VALUE.
p: AB is proper notation for “line AB”
q: centimeters are metric units.
r: 9 is prime number
a) p or q
AB is proper notation for “line AB” or centimeters
are metric units.
b)
qr:
Centimeters are metric units or 9 is a
prime number.
WRITE A COMPOUND STATEMENT FOR THE
DISJUNCTION AND FIND ITS TRUTH VALUE.
p: a diameter of a circle is twice its radius
q: a rectangle has four equal sides.
VENN DIAGRAMS:
 show
relationships between different sets of data.
 can represent conditional statements.
 is usually drawn as a circle.

Every point IN the circle belongs to that set.

Every point OUT of the circle does not.
Example:
.B
DOGS
.A
A =poodle ... a dog
B= horse ... NOT a dog
...B   dog
12
Animals
Dogs
Cats
Polygons
Triangles
Squares
Flowers
Red
Roses
Hamburgers
With
Cheese
With
Onions
Example1:
All right angles are congruent.
Congruent
Angles
Right
Angles
17
Example 2:
Every rose is a flower.
Flower
Rose
18
TO SHOW RELATIONSHIPS USING VENN DIAGRAMS:
A
B
AB
Lesson 2-2: Logic
19
THE VENN DIAGRAM SHOWS THE NUMBER OF
STUDENTS ENROLLED IN MONIQUES’ DANCE
SCHOOL FOR TAP, JAZZ AND BALLET CLASSES
Tap
Jazz
13
28
17
9
43
25
29
Ballet
a)
b)
c)
How Many students are in all three classes?
How many in tap or ballet?
How many are in jazz and ballet but not tap?
CONDITIONAL STATEMENT
Definiti A conditional statement is a
on:
statement that can be written in if-then
form.
“If _____________, then
______________.”
“if p, then q”. Symbolic Notation p → q
CONDITIONAL STATEMENT
Conditional Statements have two parts:
The hypothesis is the part of a conditional statement
that follows “if” (Usually denoted p.)
The hypothesis is the given information, or the
condition.
The conclusion is the part of an if-then statement that
follows “then” (Usually denoted q.)
The conclusion is the result of the given
information.
EXAMPLE
Write the statement “ An angle of 40° is acute.”
Hypothesis – An angle of 40° Represented by : p
Conclusion – is Acute
Represented by : q
If – Then Statement – If an angle is 40°, then the angle
is acute.
EXAMPLE
Identify the Hypothesis and Conclusion in the following
statements:
p
q
If a polynomial has six sides, then it is a hexagon.
H: A polygon has 6 sides C: it is a hexagon
1.
Tamika will advance to the next level of play if she
completes the maze in her computer game.
H: Tamika Completes the maze in her computer game.
C: She will advance to the next level of play.
2.
FORMS OF CONDITIONAL STATEMENTS
Conditional Statements:
Formed By: Given Hypothesis and Conclusion.
Symbols: p → q
Examples: If two angles have the same measure
then they are congruent.
FORMS OF CONDITIONAL STATEMENTS
Converse:
Formed By: Exchanging Hypothesis and
conclusion of the conditional.
Symbols: q → p
Examples: If two angles are congruent then they
have the same measure.
FORMS OF CONDITIONAL STATEMENTS
Inverse:
Formed By: Negating both the Hypothesis and
conclusion of the conditional.
Symbols: ~p →~q
Examples: If two angles do not have the same
measure they are not congruent.
FORMS OF CONDITIONAL STATEMENTS
Contra - positive:
Formed By: Negating both the Hypothesis and
conclusion of the Converse statement.
Symbols: ~q →~p
Examples: If two angles are not congruent then
they do not have the same measure.
Logically Equivalent Statements - are statements with
the same truth values.
Example: Write the converse, inverse and contra positive of the following statement:
Conditional: If a shape is a square, then it is a rectangle.
Converse: If a shape is a rectangle, then it is a square.
Inverse: If a shape is not a square, then it is not a
rectangle.
Contra-positive: If a shape is not a rectangle, then it is
not a square.
TRY THIS:
Example: Write the converse, inverse and contra positive of the following statement:
Conditional: If two angles form a linear pair, then
they are supplementary.
Converse:
Inverse:
Contra – positive:
ASSIGNMENTS
Classwork: Handout
Homework: pg 78-79 16-26 even and 40-44 even
DEDUCTIVE REASONING
Definition – Use facts, definitions, accepted
proportions and the laws of logic to form a logical
argument.
There are two laws of logic:
1.
The law of Detachment
2.
The law of Syllogism
LAW OF DETACHMENT
Definition – If the Hypothesis of a true conditional
statement is true, then the conclusion is also true.
So, if p → q is true and p is true, the q is true.
Symbolic Representation: p → q
p
q
Ex:
 Angles that are supplementary have measures with a
sum of 180°.
 < A and < B are Supplementary
< A and < B measures are a sum of 180°

LAW OF SYLLOGISM
Definition –
If hypothesis p, then conclusion q.
If hypothesis q , then conclusion r.
(if both above statements are true)
If hypothesis p, then conclusion r.
(Then the above is also true)
Symbolic Representation: p → q
q→r pr
Ex:
 The sun is a star
 Stars are in constant motion.
The sun is in constant motion.

Determine if statement (3) follows from statements
(1) and (2) by the Law of Detachment or the Law
of Syllogism. If it does, state which law was used.
If it does not, write invalid.









In – line skates live dangerously.
If you live dangerously, then you like to dance.
If you are an in – line skater, then you like to dance.
If you drive safely, the life you save may be your own.
Shani drives safely.
The life she saves may be her own.
If a figure is a rectangle, then its opposite sides are congruent.
AB  DC and AD  BC.
ABCD is a rectangle.
Determine if a valid conclusion can be reached from the
two true statements using the Law of Detachment or
the Law of Syllogism. If a valid conclusion is possible,
state it and the law that is used. If a valid conclusion
does not follow, write no conclusion.






If two angles are vertical, then they do not form a linear
pair.
If two angles are vertical, then they are congruent.
If you eat to live, then you live to eat.
Odina eats to live.
Cars are useful.
Useful cars are practical.
ASSIGNMENTS
Classwork: Worksheet
Homework: Pg 84-85 8-9, 24-29